This was [asked but never answered at MSE](https://math.stackexchange.com/questions/2538443/positive-sum-of-sines).

Let $f(x) = \sin(a_1x) + \sin(a_2x) + \cdots + \sin(a_nx)$, where the $a_i$'s
represent distinct positive integers.  Suppose also that $f(x)$ satisfies the inequality  $f(x) \geq 0$ on the open interval  $0 < x < \pi.$ 

In the case n=1 of a single summand it is obvious that only $f(x) = \sin x$ satisfies the condition.  For two summands, a short argument shows that only
$f(x) = \sin x + \sin(3x)$ works.  Following up on this, we define $g_n(x) = \sin x 
+ \sin(3x) + \sin(5x) + \cdots + \sin((2n-1)x)$.  Computing the sum explicitly yields $g_n(x) = \frac{\sin^2(nx)}{\sin x}$ which makes it clear that $g_n(x)$ is
nonnegative on $(0,\pi).$  

Questions: (1) Are there any other examples of $f(x)$ as above besides $g_n(x)$?
If so, can one classify them all?

(2)  The special case $f(x) = g_1(x) = \sin x$ satisfies the stronger condition of being strictly positive over $0 < x < \pi$.  Is $\sin x$ the unique
such instance of $f(x) > 0$ on $(0,\pi)$?

Thanks